Homework Help: Linearly Dependent Question

1. Oct 23, 2008

ahamdiheme

1. The problem statement, all variables and given/known data
Suppose v_1,...,v_k is a linearly dependent set. Then show that one of the vectors must be a linear combination of the others.

2. Relevant equations

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3. The attempt at a solution

I have attached an attempt at the problem. Thank you for help

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2. Oct 23, 2008

HallsofIvy

Your solution says $a_1v_1+ a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_nv_n= 0$ and then you go to $v_1= (-1/a_1)(a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_nv_n)$

That's pretty good. The only problem is you don't know that a1 is not 0! If it is you can't solve for v1. What you DO know, from the definition of "dependent", that you didn't say is that at least one of the "ai" is NOT 0. You don't know which one but you can always say "Let "k" be such that ak is not 0". Then what?

3. Oct 23, 2008

ahamdiheme

Thank you. What if i say let k:ak not equal to 0
and v1 is a linear combination of v2,...,vn iff a1=ak

4. Oct 23, 2008

Staff: Mentor

You want to show that vk is a linear combination of the rest of the vectors. IOW, that vk is a linear combination of v1, v2, ..., vk-1, vk+1, ..., vn.

5. Oct 23, 2008

ahamdiheme

ok, i think i get it